Subgroups — #13726

Order 256.

115 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄	C₂	1	4	yes	maximal
2	#2247	C₂	1	2	yes	maximal
3	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂	—	2	2	no	—
5	C₂×Q₃₂	C₂²	1	4	yes	—
6	#494	—	2	2	no	—
7	Q₆₄	—	2	8	no	—
8	C₂×C₃₂	C₂²	1	2	yes	—
9	C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆	—	2	4	no	—
11	C₂×Q₁₆	—	4	2	no	—
12	C₄⋊Q₈	—	4	2	no	—
13	Q₃₂	—	2	8	no	—
14	Q₃₂	—	4	4	no	—
15	C₂×C₁₆	D₄	1	1	yes	characteristic
16	C₂×C₁₆	C₂³	1	1	yes	characteristic
17	C₄×C₈	D₄	1	1	yes	characteristic
18	C₃₂	D₄	1	4	yes	—
19	C₂×Q₈	—	4	4	no	—
20	Q₁₆	—	4	4	no	—
21	Q₁₆	—	4	4	no	—
22	Q₁₆	—	8	4	no	—
23	C₂×C₈	D₈	1	1	yes	characteristic
24	C₂×C₈	C₂×D₄	1	1	yes	characteristic
25	C₄⋊C₄	—	8	4	no	—
26	C₄²	D₈	1	1	yes	characteristic
27	C₁₆	D₈	1	2	yes	—
28	C₁₆	C₂×D₄	1	2	yes	—
29	Q₈	—	8	4	no	—
30	Q₈	—	8	4	no	—
31	C₂×C₄	—	8	4	no	—
32	C₂×C₄	D₁₆	1	2	yes	—
33	C₂×C₄	C₂×D₈	1	1	yes	characteristic
34	C₈	D₁₆	1	2	yes	—
35	C₈	#127	1	1	yes	characteristic
36	C₈	C₂×D₈	1	1	yes	characteristic
37	C₂²	C₂×D₁₆	1	1	yes	characteristic
38	C₄	—	16	4	no	—
39	C₄	Q₆₄	1	4	yes	complemented
40	C₄	#493	1	1	yes	characteristic
41	C₄	C₂×D₁₆	1	1	yes	characteristic
42	C₂	#2246	1	1	yes	characteristic
43	C₂	C₂×Q₆₄	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
39. C₄	Q₆₄	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.